Besides, quantum metrology is a mechanism that uses the characteristic resources of quantum mechanics, such as entanglement and quantum discord, to improve the parameter estimation precision via quantum measurements beyond its classical counterpart limit. Therefore, it can be argued that quantum resources are very important in all quantum metrology protocols [1]}. In this review, our concerns include the intrinsic connections between quantum correlations and estimation theory and their roles in improving estimated parameter precision. For this purpose, this review is presented with many technical details, which could help researchers to follow and better understand the corresponding results. Further to presenting an overview of the main developments in estimation theory, we try to summarize and reformulate some of the calculations scattered in a large number of works, combined of course with our own results. To begin with, we introduce the basic concepts of classical estimation theory before discussing the quantum advantage. We then derive the classical Fisher information as a measure of information that can be related to the variance of an estimated parameter. We then show how classical Fisher information can be generalized for all possible measures to quantum Fisher information. The motivation for studying these quantities becomes clear when we introduce and derive the Cramér-Rao bound, which relates the quantum and classical Fisher information to the variance. In fact, this Cramér-Rao quantum bound is always reached in saturation in the case where only one parameter is estimated, and in this situation, the quantum correlations help us to improve the accuracy and efficiency of quantum metrology protocols. Conversely, this limit is hard to saturate in the case of multiparametric estimation due to the incompatibility between the optimal measures of the different estimated parameters. The pertinent object in the multiparametric estimation problem is given by the so-called quantum Fisher information matrix. To this end, in the second part of this review, we provide the complete techniques on the computation of this matrix, and we show that the simultaneous multi-parameter strategy is always advantageous and can provide better precision than the individual strategy in multiparameter estimation procedures. Further, we intend to focus our attention on the interaction between quantum correlations and quantum estimation theory and the characterization of quantum correlations in terms of quantum Fisher information will be discussed.
